An Algebraic Theory of Complexity for Discrete Optimization

Discrete optimization problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimized can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show...

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Bibliographic Details
Published in:SIAM journal on computing 2013-01, Vol.42 (5), p.1915-1939
Main Authors: Cohen, David A., Cooper, Martin C., Creed, Páidí, Jeavons, Peter G., Živný, Stanislav
Format: Article
Language:English
Subjects:
Algebra
Applied mathematics
Artificial Intelligence
Boolean
Cloning
Computation and Language
Computer Science
Integer programming
International conferences
Language
Logic in Computer Science
Machine Learning
Optimization
Polymorphism
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Summary:Discrete optimization problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimized can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimization problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterized. These results provide a new approach to studying the complexity of discrete optimization. We use this approach to identify certain maximal tractable subproblems of the general problem and hence derive a complete classification of complexity for the Boolean case. [PUBLICATION ABSTRACT]
ISSN:0097-5397
1095-7111
DOI:10.1137/130906398